Triangle Singularity in the Production of $T_{cc}^+(3875)$ and a Soft Pion*

The double-charm tetraquark meson $T_{cc}^+(3875)$ can be produced in high-energy proton-proton collisions by the creation of  the charm mesons $D^{*+} D^0$ at short distances followed by their binding  into  $T_{cc}^+$.  The $T_{cc}^+$ can also be produced by the creation of $D^{*+} D^{*+}$ at short distances followed by their rescattering  into $T_{cc}^+ \pi^+$.  nA charm-meson triangle singularity produces a narrow peak in the $T_{cc}^+ \pi^+$ invariant mass distribution 6.1~MeV above the threshold with a width of about 1~MeV. Beyond the peak, the differential cross section decreases with the invariant kinetic energy $E$ of $T_{cc}^+ \pi^+$ as $E^{-1/2}$.  Estimates of the fraction of $T_{cc}^+$ events accompanied by a soft $\pi^+$ and the fraction of $T_{cc}^+$ events with $T_{cc}^+\, \pi^+$ in the narrow peak from the triangle singularity suggest that they may be observable in the existing LHCb data.